The particular case of three forces is of interest. If they are not all parallel they must be concurrent, and their vector-sum must be zero. Thus three forces acting perpendicular to the sides of a triangle at the middle points will be in equilibrium provided they are proportional to the respective sides, and act all inwards or all outwards. This result is easily extended to the case of a polygon of any number of sides; it has an important application in hydrostatics. Again, suppose we have a bar AB resting with its ends on two smooth inclined planes which face each other. Let G be the centre of gravity (§ 11), and let AG = a, GB = b. Let [alpha], [beta] be the inclinations of the planes, and [theta] the angle which the bar makes with the vertical. The position of equilibrium is determined by the consideration that the reactions at A and B, which are by hypothesis normal to the planes, must meet at a point J on the vertical through G. Hence JG/a = sin ([theta] - [alpha])/sin [alpha], JG/b = sin ([theta] + [beta])/sin [beta], whence a cot [alpha] - b cot [beta] cot [theta] = ----------------------------. (6) a + b If the bar is uniform we have a = b, and cot [theta] = ½ (cot [alpha] - cot [beta]). (7) The problem of a rod suspended by strings attached to two points of it is virtually identical, the tensions of the strings taking the place of the reactions of the planes. [Illustration: FIG. 18.] Just as a system of forces is in general equivalent to a single force, so a given force can conversely be replaced by combinations of other forces, in various ways. For instance, a given force (and consequently a system of forces) can be replaced in one and only one way by three forces acting in three assigned straight lines, provided these lines be not concurrent or parallel. Thus if the three lines form a triangle ABC, and if the given force F meet BC in H, then F can be resolved into two components acting in HA, BC, respectively. And the force in HA can be resolved into two components acting in BC, CA, respectively. A simple graphical construction is indicated in fig. 19, where the dotted lines are parallel. As an example, any system of forces acting on the lamina in fig. 9 is balanced by three determinate tensions (or thrusts) in the three links, provided the directions of the latter are not concurrent. [Illustration: FIG. 19.] If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish. Hence in "trilinear" co-ordinates, with ABC as fundamental triangle, its equation is P[alpha] + Q[beta] + R[gamma] = 0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this becomes the "line at infinity," and the forces reduce to a couple. [Illustration: FIG. 20.]